L(s) = 1 | + (−0.866 − 1.5i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−1 + 1.73i)9-s − 1.73·15-s + (−1.5 − 0.866i)21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + 1.73·27-s − 29-s − 0.999i·35-s + 41-s + 1.73·43-s + (1 + 1.73i)45-s + (0.499 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.5i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−1 + 1.73i)9-s − 1.73·15-s + (−1.5 − 0.866i)21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + 1.73·27-s − 29-s − 0.999i·35-s + 41-s + 1.73·43-s + (1 + 1.73i)45-s + (0.499 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7742049344\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7742049344\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05394063168353955287826436403, −9.876490969190802781528239613115, −8.725180492350956071666244813347, −7.73817824788006451697419326899, −7.26702222242747083871665126592, −5.92209932762572805333585589650, −5.49400628669275018076183489225, −4.27225319865877781538750414638, −2.07368553047351004769500475548, −1.15755955852674574951901317778,
2.40141446604507250456079340840, 3.82425520553671718166084545484, 4.76452794896121707658351933171, 5.69922555278521477478513933344, 6.33210415928402540225887054118, 7.73259036594636184488012975004, 8.970240607895558071436510096024, 9.671962454973073291733177954578, 10.60629516383392414315482365935, 10.95796531676545423134368121845